cp's OEIS Frontend

A082184 The a(n)-th triangular number is the sum of the n-th triangular number and the smallest triangular number possible.

%I A082184 #24 Jun 05 2020 06:00:48
%S A082184 3,6,10,6,8,28,13,10,13,18,21,16,15,26,136,21,23,40,21,23,28,38,27,31,
%T A082184 28,28,61,36,38,496,53,36,43,36,61,46,41,44,106,51,53,91,45,49,58,78,
%U A082184 66,52,54,53,112,66,55,58,78,62,73,98,101,76,67,106,166,66,83,142,71
%N A082184 The a(n)-th triangular number is the sum of the n-th triangular number and the smallest triangular number possible.
%C A082184 a(n) is triangular if n+1 is triangular. Conjectures: partial maxima of sequence are at index i with value from A068195 and also a(i) - A082183(i) = 1, where i is in A068194.
%H A082184 Alois P. Heinz, <a href="/A082184/b082184.txt">Table of n, a(n) for n = 2..65536</a>
%H A082184 J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, <a href="http://arxiv.org/abs/2004.14000">Three Cousins of Recaman's Sequence</a>, arXiv:2004:14000 [math.NT], April 2020.
%p A082184 a:= proc(n) local h, j; h:= n*(n+1); for j from n+1 do
%p A082184       if issqr(1+4*(j*(j+1)-h)) then return j fi od
%p A082184     end:
%p A082184 seq(a(n), n=2..70);  # _Alois P. Heinz_, Jul 31 2019
%t A082184 a[n_] := Module[{h = n(n+1), j}, For[j = n+1, True, j++, If[IntegerQ[ Sqrt[1 + 4 (j(j+1) - h)]], Return[j]]]];
%t A082184 a /@ Range[2, 70] (* _Jean-François Alcover_, Jun 05 2020, after Maple *)
%o A082184 (PARI) for(n=2, 100, t=n*(n+1)/2; for(k=1, 10^9, u=t+k*(k+1)/2; v=floor(sqrt(2*u)); if(v*(v+1)/2==u, print1(v", "); break)))
%Y A082184 Cf. A000217, A080824, index of second term is in A082183.
%Y A082184 Partial maxima have index in A068194.
%K A082184 nonn
%O A082184 2,1
%A A082184 _Ralf Stephan_, Apr 06 2003